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A charge simulation method for the numerical conformal mapping of interior, exterior and doubly-connected domains. (English) Zbl 0818.30004
It is well known that the conformal mapping of a simply [or doubly] connected domain \(D\) onto the unit disk [or a circular ring] can be reduced to the solution of a Dirichlet problem. The author solves the latter by the ‘charge simulation method’ which has been studied extensively by several Japanese authors: Murashima, Katsurada, Okamoto and the present author. In this method the solution of the Dirichlet problem is approximated by \(G(z)= \sum^ N_{i= 1} Q_ i\log | z- \zeta_ i|\) with unknown coefficients \(Q_ i\), where the charge points \(\zeta_ i\) lie outside \(\overline D\). To determine the \(Q_ i\) it is required that \(G(z_ j)= b(z_ j)\) for \(N\) collocation points \(z_ j\in \partial D\). The error is defined by \(E_ G= \max| G(z_{j+{1\over 2}})- b(z_{j+{1\over 2}})|\), where \(z_{j+{1\over 2}}\in \partial D\) are intermediate points. The determination of \(G\) thus leads to the solution of a \(N\times N\) linear system for the \(Q_ i\). – The paper fully covers earlier work and gives a unified treatment of three types of mapping problems. The choice of the charge points is discussed, and the condition of the \(N\times N\) matrix is studied. Results of 11 numerical examples are given and compared with previous experiments. Good results are reported in all cases provided that no concave corners on \(\partial D\) occur.
Reviewer: D.Gaier (Gießen)

MSC:
30C30 Schwarz-Christoffel-type mappings
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